Representation and reconstruction of covariance operators in linear inverse problems

TL;DR

This paper presents a new framework for reconstructing functions and covariance operators in inverse problems, leveraging spatial regularization to handle ill-posedness and applying it to MEG data for functional connectivity analysis.

Contribution

It introduces a novel methodology for representing and reconstructing covariance operators in inverse problems, accommodating non-Euclidean structures and indirect noisy measurements.

Findings

Effective reconstruction of covariance operators from noisy measurements.

Application to MEG data reveals new insights into functional connectivity.

Regularization improves stability and accuracy of inverse solutions.

Abstract

We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional images or to the associated covariance operators, representing second-order information, and thus lying on a non-Euclidean space. To deal with the ill-posedness of the inverse problem, we exploit the spatial structure of the sample data by introducing a flexible regularizing term embedded in the model. Thanks to its efficiency, the proposed model is applied to MEG data, leading to a novel approach to the investigation of functional connectivity.